LMFDB - Newform orbit 18.15.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,15,Mod(17,18)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 15, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("18.17");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$N$	$=$	$18 = 2 \cdot 3^{2}$
Weight:	$k$	$=$	$15$
Character orbit:	$[\chi]$	$=$	18.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3792142673$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 44875x^{2} + 44876x + 503643366$ x^4 - 2x^3 - 44875x^2 + 44876*x + 503643366
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{7}\cdot 3^{8}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$ .

$n$	$11$
$\chi(n)$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

17.1

−149.301	−	1.41421i
150.301	−	1.41421i
150.301	+	1.41421i
−149.301	+	1.41421i

−

90.5097i

−8192.00

−

132922.i

1.63562e6

741455.i

−1.20307e7

17.2

−

90.5097i

−8192.00

141636.i

−305793.

741455.i

1.28194e7

17.3

90.5097i

−8192.00

−

141636.i

−305793.

−

741455.i

1.28194e7

17.4

90.5097i

−8192.00

132922.i

1.63562e6

−

741455.i

−1.20307e7

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	18.15.b.b	✓	4
3.b	odd	2	1	inner	18.15.b.b	✓	4
4.b	odd	2	1		144.15.e.b		4
12.b	even	2	1		144.15.e.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.15.b.b	✓	4	1.a	even	1	1	trivial
18.15.b.b	✓	4	3.b	odd	2	1	inner
144.15.e.b		4	4.b	odd	2	1
144.15.e.b		4	12.b	even	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{4} + 37728973188T_{5}^{2} + 354437717876817322500$ T5^4 + 37728973188*T5^2 + 354437717876817322500 acting on $S_{15}^{\mathrm{new}}(18, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + 8192)^{2}$ (T^2 + 8192)^2
$3$	$T^{4}$ T^4
$5$	$T^{4} + \cdots + 35\!\cdots\!00$ T^4 + 37728973188*T^2 + 354437717876817322500
$7$	$(T^{2} - 1329832 T - 500161786544)^{2}$ (T^2 - 1329832*T - 500161786544)^2
$11$	$T^{4} + \cdots + 64\!\cdots\!84$ T^4 + 1072143970894656*T^2 + 6401736089707180968250205184
$13$	$(T^{2} + \cdots - 24\!\cdots\!76)^{2}$ (T^2 - 30046336*T - 2421155104986176)^2
$17$	$T^{4} + \cdots + 21\!\cdots\!84$ T^4 + 298610596201648644*T^2 + 21887620320087675056830694566830084
$19$	$(T^{2} + \cdots - 35\!\cdots\!84)^{2}$ (T^2 + 110367392*T - 354506690023741184)^2
$23$	$T^{4} + \cdots + 18\!\cdots\!44$ T^4 + 17721255194817350976*T^2 + 18055286550022184334124099482398598144
$29$	$T^{4} + \cdots + 16\!\cdots\!64$ T^4 + 1106344771866230626116*T^2 + 168272439554821947561483078639358269311364
$31$	$(T^{2} + \cdots + 65\!\cdots\!16)^{2}$ (T^2 + 53168852792*T + 656614142473240862416)^2
$37$	$(T^{2} + \cdots + 97\!\cdots\!56)^{2}$ (T^2 + 65151669668*T + 975283163870755033156)^2
$41$	$T^{4} + \cdots + 13\!\cdots\!64$ T^4 + 73158840315557012875716*T^2 + 1334265555535559754437874368478506455286821764
$43$	$(T^{2} + \cdots - 23\!\cdots\!00)^{2}$ (T^2 - 384968187280*T - 23901675156691281272000)^2
$47$	$T^{4} + \cdots + 89\!\cdots\!00$ T^4 + 425086908378522643540800*T^2 + 897090116349519409286262022825014099216000000
$53$	$T^{4} + \cdots + 50\!\cdots\!44$ T^4 + 1028960703383742953036676*T^2 + 50393820164731953304778142988236262095920944644
$59$	$T^{4} + \cdots + 41\!\cdots\!84$ T^4 + 4099313940833855670822144*T^2 + 413432506612504667534399397495019008725880029184
$61$	$(T^{2} + \cdots + 10\!\cdots\!00)^{2}$ (T^2 - 6758379591940*T + 10676681304058450072874500)^2
$67$	$(T^{2} + \cdots + 95\!\cdots\!04)^{2}$ (T^2 + 8781899404304*T + 9551859728864961297565504)^2
$71$	$T^{4} + \cdots + 27\!\cdots\!04$ T^4 + 286170848755480055226602304*T^2 + 2715626986975023323669978627976258712032913941504
$73$	$(T^{2} + \cdots + 85\!\cdots\!36)^{2}$ (T^2 + 20287812674912*T + 85568654159913751328609536)^2
$79$	$(T^{2} + \cdots - 19\!\cdots\!44)^{2}$ (T^2 + 11008880913032*T - 198892177533121430045252144)^2
$83$	$T^{4} + \cdots + 57\!\cdots\!44$ T^4 + 401321161107559558880265024*T^2 + 577345922219773687584153283507634183198450552300544
$89$	$T^{4} + \cdots + 16\!\cdots\!44$ T^4 + 348895630398253588691328324*T^2 + 16912426630312914709499603459985975944440608072371844
$97$	$(T^{2} + \cdots - 38\!\cdots\!24)^{2}$ (T^2 - 33769679493952*T - 3839199086015747490943495424)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + 64 \beta_1 q^{2} - 8192 q^{4} + ( - \beta_{3} - 3081 \beta_1) q^{5} + ( - 5 \beta_{2} + 664916) q^{7} - 524288 \beta_1 q^{8} + (64 \beta_{2} + 394368) q^{10} + (110 \beta_{3} + 12410508 \beta_1) q^{11}+ \cdots + ( - 851092480 \beta_{3} + 45194578419648 \beta_1) q^{98}+O(q^{100})$ q + 64b1 q^2 - 8192 * q^4 + (-b3 - 3081b1) q^5 + (-5b2 + 664916) q^7 - 524288b1 q^8 + (64b2 + 394368) q^10 + (110b3 + 12410508b1) * q^11 + (-265b2 + 15023168) q^13 + (-640b3 + 42554624b1) * q^14 + 67108864 * q^16 + (190b3 + 272603169b1) * q^17 + (3080b2 - 55183696) q^19 + (8192b3 + 25239552b1) * q^20 + (-7040b2 - 1588545024) q^22 + (-18650b3 + 1073717412b1) * q^23 + (-6162b2 - 12760970969) q^25 + (-33920b3 + 961482752b1) * q^26 + (40960b2 - 5446991872) q^28 + (159875b3 + 5978338977b1) * q^29 + (-36465b2 - 26584426396) q^31 + 4294967296b1 q^32 + (-12160b2 - 34893205632) q^34 + (-634106b3 + 92178901164b1) * q^35 + (47740b2 - 32575834834) q^37 + (394240b3 - 3531756544b1) * q^38 + (-524288b2 - 3230662656) q^40 + (1392710b3 + 3599313927b1) * q^41 + (1271670b2 + 192484093640) q^43 + (-901120b3 - 101666881536b1) * q^44 + (1193600b2 - 137435828736) q^46 + (-2201010b3 - 246218864460b1) * q^47 + (-6649160b2 + 706165287807) q^49 + (-788736b3 - 816702142016b1) * q^50 + (2170880b2 - 123069792256) q^52 + (4427855b3 + 269255767113b1) * q^53 + (12749418b2 + 2149478712216) q^55 + (5242880b3 - 348607479808b1) * q^56 + (-10232000b2 - 765227389056) q^58 + (-6109100b3 + 820463915256b1) * q^59 + (-4437660b2 + 3379189795970) q^61 + (-4667520b3 - 1701403289344b1) * q^62 - 549755813888 * q^64 + (-13390238b3 + 4947771509472b1) * q^65 + (-16065930b2 - 4390949702152) q^67 + (-1556480b3 - 2233165160448b1) * q^68 + (40582784b2 - 11798899348992) q^70 + (61258090b3 + 6015258521676b1) * q^71 + (21442860b2 - 10143906337456) q^73 + (6110720b3 - 2084853429376b1) * q^74 + (-25231360b2 + 452064837632) q^76 + (-50964320b3 - 2113080472272b1) * q^77 + (-77979415b2 - 5504440456516) q^79 + (-67108864b3 - 206762409984b1) * q^80 + (-89133440b2 - 460712182656) q^82 + (-68456790b3 + 7494808273716b1) * q^83 + (273188559b2 + 5260426007058) q^85 + (162773760b3 + 12318981992960b1) * q^86 + (57671680b2 + 13013360836608) q^88 + (89881780b3 - 3331667347209b1) * q^89 + (-251318580b2 + 59929723674688) q^91 + (152780800b3 - 8795893039104b1) * q^92 + (140864640b2 + 31516014650880) q^94 + (36204736b3 - 57874123566384b1) * q^95 + (-330792850b2 + 16884839746976) q^97 + (-851092480b3 + 45194578419648b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 32768 q^{4} + 2659664 q^{7} + 1577472 q^{10} + 60092672 q^{13} + 268435456 q^{16} - 220734784 q^{19} - 6354180096 q^{22} - 51043883876 q^{25} - 21787967488 q^{28} - 106337705584 q^{31} - 139572822528 q^{34}+ \cdots + 67539358987904 q^{97}+O(q^{100})$ 4 * q - 32768 * q^4 + 2659664 * q^7 + 1577472 * q^10 + 60092672 * q^13 + 268435456 * q^16 - 220734784 * q^19 - 6354180096 * q^22 - 51043883876 * q^25 - 21787967488 * q^28 - 106337705584 * q^31 - 139572822528 * q^34 - 130303339336 * q^37 - 12922650624 * q^40 + 769936374560 * q^43 - 549743314944 * q^46 + 2824661151228 * q^49 - 492279169024 * q^52 + 8597914848864 * q^55 - 3060909556224 * q^58 + 13516759183880 * q^61 - 2199023255552 * q^64 - 17563798808608 * q^67 - 47195597395968 * q^70 - 40575625349824 * q^73 + 1808259350528 * q^76 - 22017761826064 * q^79 - 1842848730624 * q^82 + 21041704028232 * q^85 + 52053443346432 * q^88 + 239718894698752 * q^91 + 126064058603520 * q^94 + 67539358987904 * q^97

$\beta_{1}$	$=$	$( 2\nu^{3} - 3\nu^{2} - 44867\nu + 22434 ) / 89769$ (2v^3 - 3v^2 - 44867*v + 22434) / 89769
$\beta_{2}$	$=$	$( -864\nu^{3} + 1296\nu^{2} + 58162752\nu - 29081592 ) / 29923$ (-864v^3 + 1296v^2 + 58162752*v - 29081592) / 29923
$\beta_{3}$	$=$	$324\nu^{2} - 324\nu - 7269912$ 324v^2 - 324v - 7269912

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	18.15.b.b

Level	$18$
Weight	$15$
Character orbit	18.b
Analytic conductor	$22.379$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$

$\nu$	$=$	$( \beta_{2} + 1296\beta _1 + 648 ) / 1296$ (b2 + 1296*b1 + 648) / 1296
$\nu^{2}$	$=$	$( 4\beta_{3} + \beta_{2} + 1296\beta _1 + 29080296 ) / 1296$ (4b3 + b2 + 1296b1 + 29080296) / 1296
$\nu^{3}$	$=$	$( 6\beta_{3} + 22435\beta_{2} + 87246072\beta _1 + 43620120 ) / 1296$ (6b3 + 22435b2 + 87246072*b1 + 43620120) / 1296

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format: